Shape Representation and Classification Using the Poisson Equation
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#Distance Transform

## Eikonal equation:

* How to derive it?

[What are the eikonal equations and how are they derived?](http://www.quora.com/What-are-the-eikonal-equations-and-how-are-they-derived)

$\frac{n^2}{c^2}\frac{\partial^2 u}{\partial t^2} - \nabla^2 u = 0$

... =>

$\left \| \nabla u \right \|^2 = 1$

# Poisson Equation

## How to derive it?

TODO

$\Delta u = f$

# Discrete Poisson equation

Using the finite difference numerical method to discretize the 2 dimensional Poisson equation (assuming a uniform spatial discretization, $\Delta x=\Delta y$) on an m × n grid:

$({\nabla}^2 u )_{ij} = \frac{1}{\Delta x^2} (u_{i+1,j} + u_{i-1,j} + u_{i,j+1} + u_{i,j-1} - 4 u_{ij})$

(considering $\frac{\partial^2 u}{\partial x^2} = \frac{\frac{u_{i+1,j} - u_{i,j}}{\Delta x} - \frac{u_{i1,j} - u_{i-1,j}}{\Delta x}}{\Delta x}$)




